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C2×C3⋊(C5⋊(ΓL(1,𝔽8)⋊D19)) #37561488

order23940
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3⋊(C5⋊(ΓL(1,𝔽8)⋊D19))
unambiguous semidirect product(C2×C3)⋊(C5⋊(ΓL(1,𝔽8)⋊D19))
unambiguous semidirect product(C2×C5)⋊(C3⋊(ΓL(1,𝔽8)⋊D19))
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊D285

Splittings

Fitting, socleC2×C3×C5×C7×C19 ⋊ C2×C3
derivedC3×C5×C7×C19 ⋊ C22×C3
HallC3×C5×C19×ΓL(1,𝔽8) ⋊ C22
HallC3×C19×ΓL(1,𝔽8) ⋊ C2×D5
HallC5×C7×C19 ⋊ C2×C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×D19
HallC7×C19 ⋊ C2×C3×D15
HallC5×C19 ⋊ C2×C3⋊AGL(1,𝔽7)
HallC3×ΓL(1,𝔽8) ⋊ C2×D95
HallC5×C7 ⋊ C2×C3×D57
SylowC19 ⋊ C2×C3⋊(C5⋊AGL(1,𝔽7))
SylowC7 ⋊ C2×C3×D285
SylowC5 ⋊ C2×C3⋊(ΓL(1,𝔽8)⋊D19)

Subgroups

subgroupG/H
center, hypercenterC2C3⋊(C5⋊(ΓL(1,𝔽8)⋊D19))
commutator, cosocle, hypocenterC3×C5×C7×C19C22×C3
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C19C2×C3
LCSG ▷ C3×C5×C7×C19
UCSC1 ◁ C2

176 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3990

Conjugacy classes & character table

1434 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23940, 620);